Quantitative Magnetic Resonance Imaging: From Fingerprinting to Integrated Physics-Based Models

Abstract

Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters e.g., relaxation times T1, T2, or proton density ρ. Recently in [Ma et al., Nature, 2013], Magnetic Resonance Fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two step procedure: (i) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a pre-computed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into a perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is non-convex and that the accuracy of the MRF-type algorithms is limited by the "discretization size" of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two step method, our model is dictionary-free and is rather governed by a single non-linear equation, which is studied analytically. This non-linear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples for which also improvement over MRF and its variants is documented.Here m, yielding m = ρm, is the macroscopic magnetization of (Hydrogen) proton of some unitary density in the tissue under an external magnetic field B, and the relaxation rates T 1 and T 2 are associated model parameters. Further, m 0 represents an initial state. System (1.1) is instrumental in our quantification process established below and will be further described in Section 2.1.Although qMRI techniques are still in their infancy, several interesting ideas and methods have already been conceived. Early approaches [25] are based on a set of spin echo or inversion recovery images that are reconstructed from k-space data with respect to various repetition times (T R) and echo times (T E). In that context, acquisitions are designed for each parameter individually. The overall technique is often referred to as parametric mapping method and consists of two steps: (i) reconstruct a sequence of images as in qualitative MRI, and (ii) for each pixel of those images fit its intensity to an ansatz curve characterized by the magnetic parameter associated to the tissue imaged at that pixel. Based on this idea, many improvements have been suggested in the literature; see for instance [17]. The associated approaches aim to simplify the physical model and handle tissue parameters separately, as these are considered to be time consuming for the patient.Another line of research, initiated by Ma et al. in [27] and named Magnetic Resonance Fingerprinting (MRF), has recently gained considerable attention. First, in an offline phase, it builds a database (dictio...

Optimization with learning-informed differential equation constraints and its applications

“…For the small size training data, we generate parameter values for (T 1 , T 2 ) from For the SQP we consider the image domain to be [0, 1]×[0, 1], thus the spatial discretization size is h = 1/180. We compare the results of the learning-based method with results from the algorithm proposed in our previous work [20]. The initialization to the SQP algorithm and also the algorithm in [20] is done by using the so-called BLIP algorithm of [18] with a dictionary resulting from the small size D 1 × D 2 .…”

“…Fixing the external magnetic field B according to an excitation protocol with a specific sequence of frequency pulses (cf., e.g., [20]) and associated echo times {t i } L i=1 we have u → {y(t i )} L i=1 yielding the solution map Π :…”

“…The second example comes from quantitative magnetic resonance imaging -qMRI. In this context, one integrates a mathematical model of the acquisition physics (the Bloch equations [20]) into the associated image reconstruction task in order to relate qualitative information (such as the net magnetization y = ρm) with objective, tissue dependent quantitative information (such as T 1 and T 2 , the longitudinal and the transverse relaxation times, respectively, or the proton spin density ρ). This model is then used to obtain quantitative reconstructions from noisy subsampled measurement data g in k-space by a variational approach.…”

“…The provision of such quantitative reconstructions is highly important, e.g., for subsequent automated image classification procedures to identify tissue anomalies. Moreover, in [20] it is demonstrated that such an integrated physicsbased approach is superior to the state-of-the-art technique of magnetic resonance fingerprinting (MRF) [40] and its improved variants [18,42]. Specifically in MRI, acquisition data are obtained at different pre-specified times (read-out times) t 1 , .…”

“…Crucial to this approach is the fact that, at least for specific variations of the external magnetic field B, explicit formulas for the solution map of the Bloch equations are available. For instance, in [18] and [20] Inversion Recovery balanced Steady-State Free Precession (IR-bSSFP) [49] is used which involves certain flip angle sequence patterns that characterize the external magnetic field B. These flip angle patterns allow for a simple approximation of the solutions of the Bloch equations at the read-out times through a recurrence formula.…”